Selecta Mathematica

, Volume 23, Issue 1, pp 389–423 | Cite as

Derived categories of cyclic covers and their branch divisors

  • Alexander KuznetsovEmail author
  • Alexander Perry


Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover \(X \rightarrow Y\) ramified over a divisor \(Z \subset Y\). We construct semiorthogonal decompositions of \(\mathrm {D^b}(X)\) and \(\mathrm {D^b}(Z)\) with distinguished components \({\mathcal {A}}_X\) and \({\mathcal {A}}_Z\) and prove the equivariant category of \({\mathcal {A}}_X\) (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into \(n-1\) copies of \({\mathcal {A}}_Z\). As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

Mathematics Subject Classification




A.K. is grateful to Alexey Elagin for his clarifications concerning equivariant categories. A.P. thanks Joe Harris and Johan de Jong for useful conversations related to this work.


  1. 1.
    Addington, N., Thomas, R.: Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163(10), 1885–1927 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bondal, A.: Representations of associative algebras and coherent sheaves. Izv. Akad. Nauk SSSR Ser. Mat. 53(1), 25–44 (1989)MathSciNetGoogle Scholar
  3. 3.
    Bondal, A., Kapranov, M.: Representable functors, Serre functors, and mutations. Math. USSR Izv. 35(3), 519 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bondal, A., Orlov, D.: Semiorthogonal Decompositions for Algebraic Varieties, arXiv preprint. arXiv:alg-geom/9506012 (1995)
  5. 5.
    Bridgeland, T., King, A., Reid, M.: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001). (electronic) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Collins, J., Polishchuk, A.: Gluing stability conditions. Adv. Theor. Math. Phys. 14(2), 563–607 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Debarre, O., Kuznetsov, A.: Gushel–Mukai Varieties: Classification and Birationalities, arXiv preprint. arXiv:1510.05448v1 (2015)
  8. 8.
    Elagin, A.: Cohomological descent theory for a morphism of stacks and for equivariant derived categories. Sb. Math. 202(4), 495 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Elagin, A.: Descent theory for semiorthogonal decompositions. Sb. Math. 203(5), 645–676 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Elagin, A.: On Equivariant Triangulated Categories, arXiv preprint. arXiv:1403.7027v2 (2015)
  11. 11.
    Fonarev, A.: Minimal Lefschetz decompositions of the derived categories for Grassmannians. Izv. Math. 77(5), 1044 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gushel’, N.P.: On Fano varieties of genus 6. Izv. Math. 21(3), 445–459 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ingalls, C., Kuznetsov, A.: On nodal Enriques surfaces and quartic double solids. Math. Ann. 361(1–2), 107–133 (2014)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ishii, A., Ueda, K.:The special McKay correspondence and exceptional collections. Tohoku Math. J. (2) 67(4), 585–609 (2015)Google Scholar
  15. 15.
    Kuznetsov, A.: Derived categories of cubic and \(V_{14}\) threefolds. In: Trudy Matematicheskogo Instituta im. V.A. Steklova, vol. 246, pp. 183–207 (2004)Google Scholar
  16. 16.
    Kuznetsov, A.: Hyperplane sections and derived categories. Izv. Math. 70(3), 447 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kuznetsov, A.: Hochschild Homology and Semiorthogonal Decompositions, arXiv preprint. arXiv:0904.4330v1 (2009)
  18. 18.
    Kuznetsov, A.: Derived categories of cubic fourfolds. In: Cohomological and Geometric Approaches to Rationality Problems. Springer, Berlin, pp. 219–243 (2010)Google Scholar
  19. 19.
    Kuznetsov, A.: Base change for semiorthogonal decompositions. Compos. Math. 147(3), 852–876 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kuznetsov, A.: Semiorthogonal decompositions in algebraic geometry. In: Proceedings of the International Congress of Mathematicians, vol. II (Seoul, 2014), pp. 635–660 (2014)Google Scholar
  21. 21.
    Kuznetsov, A.: Calabi–Yau and Fractional Calabi–Yau Categories, arXiv preprint. arXiv:1509.07657 (2015)
  22. 22.
    Kuznetsov, A., Lunts, V.A.: Categorical resolutions of irrational singularities. Int. Math. Res. Not. 13, 4536–4625 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kuznetsov, A., Perry, A.: Derived Categories of Gushel–Mukai Varieties, preprintGoogle Scholar
  24. 24.
    Mukai, S.: Biregular classification of Fano \(3\)-folds and Fano manifolds of coindex \(3\). Proc. Natl. Acad. Sci. USA 86(9), 3000–3002 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of Algebraic GeometrySteklov Mathematical InstituteMoscowRussian Federation
  2. 2.The Poncelet LaboratoryIndependent University of MoscowMoscowRussian Federation
  3. 3.Laboratory of Algebraic Geometry and its ApplicationsNational Research University Higher School of EconomicsMoscowRussian Federation
  4. 4.Department of MathematicsHarvard UniversityCambridgeUSA

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